A376381 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*(exp(x) - 1))^2 ).

1, 0, 4, 6, 272, 1570, 63912, 792554, 33262784, 684763650, 30981768680, 915838324522, 45524048263872, 1765020653500130, 97096528136899592, 4651295721203951850, 283478019364268181632, 16107548441248677913858, 1084981357752210351649512, 71056829948555342150405354, 5267564532376249471978526720

Offset: 0

Author

Seiichi Manyama, Sep 22 2024

Keywords

• nonn

Links

• Index entries for reversions of series

Crossrefs

Cf. A371271, A375660.

Programs

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x*(exp(x)-1))^2)/x))

PARI

a(n) = 2*n!*sum(k=0, n\2, (2*n+k+1)!*stirling(n-k, k, 2)/(n-k)!)/(2*n+2)!;

Formula

E.g.f. A(x) satisfies A(x) = 1/(1 - x*A(x) * (exp(x*A(x)) - 1))^2.

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A371271.

a(n) = (2 * n!/(2*n+2)!) * Sum_{k=0..floor(n/2)} (2*n+k+1)! * Stirling2(n-k,k)/(n-k)!.